Asymptotics of n-universal lattices over number fields

2025/10/30 by Park, Dayoon, Visser, Robin, Yatsyna, Pavlo +1
#11E12 (Primary) 11R80 #11N45 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.26652

Abstract

We prove an explicit asymptotic formula for the logarithm of the minimal ranks of n-universal lattices over the ring of integers of totally real number fields. We also show that, for any constant C > 0 and n ≥ 3, there are only finitely many totally real fields with an n-universal lattice of rank at most C, with all such fields being effectively computable. Similarly, for any n ≥ 3, we show that there are only finitely many totally real fields admitting an n-universal criterion set of size at most C, with all such fields likewise being effectively computable.

Citations

• Additively indecomposable quadratic forms over totally real number fields
• Kitaoka's Conjecture for quadratic fields
• Universality criterion sets for quadratic forms over number fields
• Real quadratic fields with a universal quadratic form of given rank have density zero
• Can we recover an integral quadratic form by representing all its subforms?
• On Kitaoka's conjecture and lifting problem for universal quadratic forms
• On the size and structure of finite linear groups
• Minimal S-universality criteria may vary in size
• Bounds for the orders of the finite subgroups of G(k)
• Orders of Finite Groups of Matrices

Related

• Extending bounds on minimal ranks of universal quadratic lattices to larger number fields
• A lower bound for the rank of a universal quadratic form with integer\n coefficients in a totally real number field
• The distribution of lattices arising from orders in low degree number fields
• Heuristics for the asymptotics of the number of S_n-number fields
• Counting Co-Cyclic Lattices